Linear Spaces

Fields

Definition: A field is a set FF of scalars with two operations, addition and multiplication, such that the following axioms hold:

Addition

Multiplication

Distributivity

At least two elements must exist in a field, additive identity 0F0_F and multiplicative identity 1F1_F.

Field examples include R\mathbb{R}, C\mathbb{C}, Q\mathbb{Q}, Zp\mathbb{Z}_p where pp is a prime number. R\mathbb{R} are the real numbers, C\mathbb{C} are the complex numbers, Q\mathbb{Q} are the rational numbers, Zp\mathbb{Z}_p are the integers modulo pp.

In a sense, what I understand from the fields are, mathematical concepts that are used to define the operations on the elements of the vector spaces.

Linear Spaces

Fn\mathbb{F}^n

Definition: The set of all lists of length nn with elements from a field F\mathbb{F} is denoted by Fn\mathbb{F}^n.

Fn={(x1,x2,…,xn)∣xi∈F,i=1,2,…,n}\mathbb{F}^n = \{(x_1, x_2, \ldots, x_n) \mid x_i \in \mathbb{F}, i=1,2,\ldots,n\}

Addition and scalar multiplication are defined as follows:

(x1,x2,…,xn)+(y1,y2,…,yn)=(x1+y1,x2+y2,…,xn+yn)(x_1, x_2, \ldots, x_n) + (y_1, y_2, \ldots, y_n) = (x_1+y_1, x_2+y_2, \ldots, x_n+y_n)
α(x1,x2,…,xn)=(αx1,αx2,…,αxn)\alpha(x_1, x_2, \ldots, x_n) = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n)

#EE501 - Linear Systems Theory at METU